1. Field of the Invention
This invention relates in general to methods for space travel, and in particular, to methods for an object, such as a satellite, space craft, and the like, to change inclinations using, for example, weak stability boundaries (WSBs) to be placed in orbit around the earth, moon, and/or other planets.
2. Background of the Related Art
The study of motion of objects, including celestial objects, originated, in part, with Newtonian mechanics. During the eighteenth and nineteenth centuries, Newtonian mechanics, using a law of motion described by acceleration provided an orderly and useful framework to solve most of the celestial mechanical problems of interest for that time. In order to specify the initial state of a Newtonian system, the velocities and positions of each particle must be specified.
However, in the mid-nineteenth century, Hamilton recast the formulation of dynamical systems by introducing the so-called Hamiltonian function, H, which represents the total energy of the system expressed in terms of the position and momentum, which is a first-order differential equation description. This first order aspect of the Hamiltonian, which represents a universal formalism for modeling dynamical systems in physics, implies a determinism for classical systems, as well as a link to quantum mechanics.
By the early 1900s, Poincare understood that the classical Newtonian three-body problem gave rise to a complicated set of dynamics that was very sensitive to dependence on initial conditions, which today is referred to as "chaos theory." The origin of chaotic motion can be traced back to classical (Hamiltonian) mechanics which is the foundation of (modern) classical physics. In particular, it was nonintegrable Hamiltonian mechanics and the associated nonlinear problems which posed both the dilemma and ultimately the insight into the occurrence of randomness and unpredictability in apparently completely deterministic systems.
The advent of the computer provided the tools which were hitherto lacking to earlier researchers, such as Poincare, and which relegated the nonintegrable Hamiltonian mechanics from the mainstream of physics research. With the development of computational methodology combined with deep intuitive insights, the early 1960s gave rise to the formulation of the KAM theorem, named after A. N. Kolmogorov, V. l. Arnold, and J. Moser, that provided the conditions for randomness and unpredictability for nearly nonintegrable Hamiltonian systems.
Within the framework of current thinking, almost synonymous with certain classes of nonlinear problems is the so-called chaotic behavior. Chaos is not just simply disorder, but rather an order without periodicity. An interesting and revealing aspect of chaotic behavior is that it can appear random when the generating algorithms are finite, as described by the so-called logistic equations.
Chaotic motion is important for astrophysical (orbital) problems in particular, simply because very often within generally chaotic domains, patterns of ordered motion can be interspersed with chaotic activity at smaller scales. Because of the scale characteristics, the key element is to achieve sufficiently high resolving power in the numerical computation in order to describe precisely the quantitative behavior that can reveal certain types of chaotic activity. Such precision is required because instead of the much more familiar spatial or temporal periodicity, a type of scale invariance manifests itself. This scale invariance, discovered by Feigenbaum for one-dimensional mappings, provided for the possibility of analyzing renormalization group considerations within chaotic transitions.
Insights into stochastic mechanics have also been derived from related developments in nonlinear analysis, such as the relationship between nonlinear dynamics and modern ergodic theory. For example, if time averages along a trajectory on an energy surface are equal to the ensemble averages over the entire energy surface, a system is said to be ergodic on its energy surface. In the case of classical systems, randomness is closely related to ergodicity. When characterizing attractors in dissipative systems, similarities to ergodic behavior are encountered.
An example of a system's inherent randomness is the work of E. N. Lorenz on thermal convection, which demonstrated that completely deterministic systems of three ordinary differential equations underwent irregular fluctuations. Such bounded, nonperiodic solutions which are unstable can introduce turbulence, and hence the appellation "chaos," which connotes the apparent random motion of some mappings. One test that can be used to distinguish chaos from true randomness is through invocation of algorithmic complexity; a random sequence of zeros and ones can only be reproduced by copying the entire sequence, i.e., periodicity is of no assistance.
The Hamiltonian formulation seeks to describe motion in terms of first-order equations of motion. The usefulness of the Hamiltonian viewpoint lies in providing a framework for the theoretical extensions into many physical models, foremost among which is celestial mechanics. Hamiltonian equations hold for both special and general relativity. Furthermore, within classical mechanics it forms the basis for further development, such as the familiar Hamilton-Jacobi method and, of even greater extension, the basis for perturbation methods. This latter aspect of Hamiltonian theory will provide a starting point for the analytical discussions to follow in this brief outline.
As already mentioned, the Hamiltonian formulation basically seeks to describe motion in terms of first-order equations of motion. Generally, the motion of an integrable Hamilton system with N degrees of freedom is periodic and confined to the N-torus as shown in FIG. 1. FIG. 1 depicts an integrable system with two degrees of freedom on a torus, and a closed orbit of a trajectory. The KAM tori are concentric versions of the single torus. Hamiltonian systems for which N=1 are all integrable, while the vast majority of systems with N greater than or equal to 2 become nonintegrable.
An integral of motion which makes it possible to reduce the order of a set of equations, is called the first integral. To integrate a set of differential equations of the order 2N, that same number of integrals are generally required, except in the case of the Hamiltonian equations of motion, where N integrals are sufficient. This also can be expressed in terms of the Liouville theorem, which states that any region of phase space must remain constant under any (integrable) Hamiltonian formalism. The phase space region can change its shape, but not its phase space volume. Therefore, for any conservative dynamical system, such as planetary motion or pendula that does not have an attracting point, the phase space must remain constant.
Another outcome of the Hamiltonian formulation, which started out as a formulation for regular motion, is the implication of the existence of irregular and chaotic trajectories. Poincare realized that nonintegrable, classical, three-body systems could lead to chaotic trajectories. Chaotic behavior is due neither to a large number of degrees of freedom nor to any initial numerical imprecision. Chaotic behavior arises from a nonlinearity in the Hamiltonian equations with initially close trajectories that separate exponentially fast into a bounded region of phase space. Since initial conditions can only be measured with a finite accuracy and the errors propagate at an exponential rate, the long range behavior of these systems cannot be predicted.
The effects of perturbations in establishing regions of nonintegrability can be described for a weak perturbation using the KAM theorem. The KAM theorem, originally stated by Kolmogorov, and rigorously proven by Arnold and Moser, analyzed perturbative solutions to the classical many-body problem. The KAM theorem states that provided the perturbation is small, the perturbation is confined to an N-torus except for a negligible set of initial conditions which may lead to a wandering motion on the energy surface. This wandering motion is chaotic, implying a great sensitivity to initial conditions.
The N-tori, in this case, are known as KAM surfaces. When observed as plane sections they are often called KAM curves as illustrated in FIG. 2. These surfaces and curves may be slightly distorted (perturbed). That is, for a sufficiently small conservative Hamiltonian perturbation, most of the nonresonant invariant tori will not vanish, but will undergo a slight deformation, such that in the perturbed system phase space there are also invariant tori, filled by phase curves, which are conditionally periodic.
FIG. 2 illustrates a set of KAM invariant tori on the surface of which lie as elliptic integrable solutions. The nonintegrable solutions, irregular paths, which are hyperbolic in nature lie in between the invariant tori in so-called resonant zones, which are also sometimes referred to as stochastic zones.
The KAM results were extended through the results of J. Mather. KAM theory treats motions and related orbits that are very close to being well behaved and stable. Since KAM theory is basically a perturbation analysis, by its very nature the perturbation constant must be very small. Strong departures from the original operator through the perturbation parameter will invalidate the use of the original eigenfunctions used to generate the set of perturbed eigenfunctions. Mather's work analyzes unstable motions which are far from being well behaved. The perturbation can be relatively strong, and entirely new eigenfunctions (solutions) can be generated.
The practical importance of Mather's work for planetary orbit, escape, and capture is that the dynamics are applicable to those regions in phase space (i.e., Mather regions) associated with three- and four-body problems. Mather proved that for chaotic regions in lower (two) dimensions for any conservative Hamiltonian System, there exists or remains elliptical orbits which are unstable. In terms of NEO (near-Earth object) issues, KAM and Mather regions are important for describing both the orbital motions of comets, as well as for planning fuel conserving ballistic (flyby, rendezvous, and interception) trajectories to comets and other NEOs. The above discussion is a summary of the article by John L. Remo, entitled "NEO Orbits and Nonlinear Dynamics: A Brief Overview and Interpretations," 822 Annals of the New York Academy of Sciences 176-194 (1997), incorporated herein by reference, including the references cited therein.
Since the first lunar missions in the 1960s, the moon has been the object of interest of both scientific research and potential commercial development. During the 1980s, several lunar missions were launched by national space agencies. Interest in the moon is increasing with the advent of the multi-national space station making it possible to stage lunar missions from low earth orbit. However, continued interest in the moon and the feasibility of a lunar base will depend, in part, on the ability to schedule frequent and economical lunar missions.
A typical lunar mission comprises the following steps. Initially, a spacecraft is launched from earth or low earth orbit with sufficient impulse per unit mass, or change in velocity, to place the spacecraft into an earth-to-moon orbit. Generally, this orbit is a substantially elliptic earth-relative orbit having an apogee selected to nearly match the radius of the moon's earth-relative orbit.
As the spacecraft approaches the moon, a change in velocity is provided to transfer the spacecraft from the earth-to-moon orbit to a moon-relative orbit. An additional change in velocity may then be provided to transfer the spacecraft from the moon-relative orbit to the moon's surface if a moon landing is planned. When a return trip to the earth is desired, another change in velocity is provided which is sufficient to insert the spacecraft into a moon-to-earth orbit, for example, an orbit similar to the earth-to-moon orbit. Finally, as the spacecraft approaches the earth, a change in velocity is required to transfer the spacecraft from the moon-to-earth orbit to a low earth orbit or an earth return trajectory.
FIG. 3 is an illustration of an orbital system in accordance with a conventional lunar mission in a non-rotating coordinate system wherein the X-axis 10 and Y-axis 12 lay in the plane defined by the moon's earth-relative orbit 36, and the Z-axis 18 is normal thereto. In a typical lunar mission, a spacecraft is launched from earth 16 or low earth orbit 20, not necessarily circular, and provided with sufficient velocity to place the spacecraft into an earth-to-moon orbit 22.
Near the moon 14, a change in velocity is provided to reduce the spacecraft's moon-relative energy and transfer the spacecraft into a moon-relative orbit 24 which is not necessarily circular. An additional change in velocity is then provided to transfer the spacecraft from the moon-relative orbit 24 to the moon 14 by way of the moon landing trajectory 25. When an earth-return is desired, a change in velocity sufficient to place the spacecraft into a moon-to-earth orbit 26 is provided either directly from the moon's surface or through multiple impulses as in the descent. Finally, near the earth 16, a change in velocity is provided to reduce the spacecraft's earth-relative energy and return the spacecraft to low earth orbit 20 or to earth 16 via the earth-return trajectory 27.
FIG. 4 is an illustration of another conventional orbital system, described in U.S. Pat. No. 5,158,249 to Uphoff, incorporated herein by reference, including the references cited therein. The orbital system 28 comprises a plurality of earth-relative orbits, where transfer therebetween is accomplished by using the moon's gravitational field. The moon's gravitation field is used by targeting, through relatively small mid-orbit changes in velocity, for lunar swingby conditions which yield the desired orbit.
Although the earth-relative orbits in the orbital system 28 may be selected so that they all have the same Jacobian constant, thus indicating that the transfers therebetween can be achieved with no propellant-supplied change in velocity in the nominal case, relatively small propellant-supplied changes in velocity may be required. Propellant-supplied changes in velocity may be required to correct for targeting errors at previous lunar swingbys, to choose between alternative orbits achievable at a given swingby, and to account for changes in Jacobian constant due to the eccentricity of the moon's earth-relative orbit 36.
In FIG. 4, a spacecraft is launched from earth 16 or low earth orbit into an earth-to-moon orbit 22. The earth-to-moon orbit 22 may comprise, for example, a near minimal energy earth-to-moon trajectory, for example, an orbit having an apogee distance that nearly matches the moon's earth-relative orbit 36 radius. The spacecraft encounters the moon's sphere of gravitational influence 30 and uses the moon's gravitational field to transfer to a first earth-relative orbit 32.
The first earth-relative orbit 32 comprises, for example, approximately one-half revolution of a substantially one lunar month near circular orbit which has a semi-major axis and eccentricity substantially the same as the moon's earth-relative orbit 36, which is inclined approximately 46.3 degrees relative to the plane defined by the moon's earth-relative orbit 36, and which originates and terminates within the moon's sphere of influence 30. Because the first earth-relative orbit 32 and a typical near minimum energy earth-to-moon orbit 22 have the same Jacobian constant, the transfer can be accomplished by using the moon's gravitational field.
FIG. 5 is an illustration of another orbital system where, for example, satellites orbit the earth. A central station SC is situated at the center of a spherical triangle-shaped covering zone Z. Two geosynchronous satellites S-A and S-B have elliptical orbits with identical parameters. These parameters may be, for example, the following:
apogee situated at about 50,543.4 km, PA1 perigee situated at about 21,028.6 km, PA1 meniscal axis of 42,164 km, PA1 inclination of 63 degrees, PA1 perigee argument 270, PA1 orbit eccentricity 0.35.
Each satellite includes an antenna or antennae 11 and 11a; each antenna is orientated towards the central station throughout the period when the satellite moves above the covering zone. The central station includes one connection station and one control station. FIG. 5 also shows a mobile unit M (which is situated inside zone Z, but which is shown above the latter for the sake of more clarity). This mobile unit is equipped with an antenna 14a whose axis continuously points substantially towards the zenith.
In order to station such satellites, a large number of strategies are possible. One exemplary strategy is described with reference to FIG. 6. This strategy uses the ARIANE IV rocket and requires three pulses. At the time of launching, the satellite is accompanied by an ordinary geostationary satellite. The two satellites are placed on the standard transfer orbit of the ARIANE IV rocket, this orbit being situated within a quasi-equatorial plane (inclination of 7 degrees) with a perigee at 200 km, an apogee at 35,975 km and a perigee argument of 178 degrees. The orbit is marked as OST on FIG. 6.
Close to the perigee, a satellite rocket is ignited for a first pulse suitable for raising the apogee to 98,000 km, the orbit remaining within the same plane, orbit 01. This pulse may be broken down into two or three pulses. Close to the apogee of the orbit 01, a new pulse is sent to the satellite to change the plane of its orbit. The inclination of this plane is close to that of the plane of the definitive orbit, namely 63 degrees. This thrust is the largest and may be broken down into two or three thrusts. The orbit then becomes 02.
Finally, at an appropriate point of this orbit, a third thrust is sent to the satellite so as to provide it with a definitive orbit. If this strategy is satisfactory in certain respects, it nevertheless does constitute a drawback. In fact, it requires that the orbital plane be tilted when passing from the orbit 01 to the orbit 02, this resulting in a considerable consumption of propellant.
FIG. 7 is an illustration of another conventional lunar gravitational assistance transfer principle. In FIG. 7, the satellite is firstly transferred onto a standard orbit 01 situated inside a quasi-equatorial plane, which, in practice, is the orbit OST of FIG. 6, known as a Geostationary Transfer Orbit (GTO) orbit. At T1, the satellite is transferred onto a circumlunar orbit 02, still situated in the quasi-equatorial plane.
In practice, an extremely elliptic orbit is selected whose major axis is close to twice the Earth/Moon distance, namely about 768,800 km. The satellite penetrates into the sphere of influence SI of the moon and leaves this sphere on a trajectory 03 whose plane is highly inclined with respect to the equatorial plane. At T2, the satellite is injected onto the definitive orbit 04 inside the same plane as the orbit 03. The above described orbital system is described in detail in U.S. Pat. No. 5,507,454 to Dulck, incorporated herein by reference, including the references cited therein.
Dulck attempts to minimize the thrusters needed, where the standard technique of lunar gravity assist is used. The satellite is first brought to a neighborhood of the moon by a Hohmann transfer. It then flies by the moon in just the right directions and velocities, where it is broken up into two or more maneuvers. This method works, but the size of this maneuver restricts the applications of the method to ellipses whose eccentricities are sufficiently large. This is because to have a savings with this large maneuver, the final maneuver needs to be sufficiently small.
I have determined that all of the above orbital systems and/or methods suffer from the requirement of substantial fuel expenditure for maneuvers, and are therefore, not sufficiently efficient. I have also determined that the above methods focus on orbital systems that concentrate on the relationship between the earth and the moon, and do not consider possible effects and/or uses beyond this two-body problem.
Accordingly, it is desirable to provide an orbital system and/or method that furnishes efficient use of fuel or propellant. It is also desirable to provide an orbital system and/or method that it not substantially dependent on significant thrusting or propelling forces.
It is also desirable to provide an orbital system and/or method that considers the effects of lunar capture and/or earth capture as more than merely a two body problem. It is also desirable to provide an orbital system and/or method that may be implemented on a computer system that is either onboard the spacecraft or satellite, or located in a central controlling area.
It is also desirable to provide an orbital system and/or method that allows a spacecraft to make repeated close approaches to both the earth and moon. It is also desirable to provide an orbital system and/or method that is sustainable with relatively low propellant requirements, thereby providing an efficient method for cislunar travel.
It is also desirable to provide an orbital system and/or method that does not require large propellant supplied changes in velocity. It is also desirable to provide an orbital system and/or method that renders practical massive spacecraft components. It is also desirable to provide an orbital system and/or method that may be used for manned and unmanned missions.
It is also desirable to provide an orbital system and/or method that allows a spacecraft or satellite to make repeated close approaches at various inclinations to both the earth and moon.
It is also desirable to provide an orbital system and/or method that allows a spacecraft or satellite to make inclination changes with respect to, for example, the earth and/or moon.